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A metric approach to zero-free regions for $L$ -functions

This paper presents a new proof of zero-free regions for $L(s, \pi)$ and $L(s, \pi \times \pi')$ under self-duality assumptions by combining Granville and Soundararajan's "pretentious" multiplicative function theory with Lichtman and Pascadi's concept of positive semi-definite families of automorphic representations.

Original authors: Nawapan Wattanawanichkul

Published 2026-06-05

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Nawapan Wattanawanichkul

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Hunting for "Ghost" Numbers

Imagine the world of numbers as a vast, infinite landscape. In this landscape, there are special functions called L-functions. You can think of these functions as complex machines that take a number (let's call it $s$ ) and spit out a result.

Mathematicians are obsessed with finding the "zeros" of these machines—specifically, the inputs where the machine outputs zero. Why? Because these zeros hold the secrets to how prime numbers are distributed. It's like trying to find the hidden pattern in a chaotic storm.

The most famous of these machines is the Riemann Zeta function. In 1896, mathematicians proved that this machine never outputs zero if you feed it a number with a specific "real part" equal to 1. They also found a narrow "safe zone" just to the left of that line where the machine is also guaranteed to be non-zero. This safe zone is called a zero-free region.

This paper is about proving that similar safe zones exist for a whole family of more complex machines (called Rankin–Selberg L-functions) that are built from "automorphic representations" (which are like advanced, multi-dimensional versions of prime number patterns).

The Old Way: The "3-4-1" Trick

For over a century, mathematicians have used a clever trick to prove these safe zones exist. It's often called the "3-4-1 argument."

Imagine you are trying to prove that a specific spot in a forest is safe from wolves.

You know that if a wolf is at spot A, it must be howling.
You also know that if a wolf is at spot B, it must be growling.
The trick involves combining these sounds in a specific ratio (3 parts howl, 4 parts growl, 1 part silence) to create a "super-sound" that is always positive.
If you assume a wolf (a zero) is hiding in your safe zone, this super-sound would have to be negative, which is impossible. Therefore, no wolf can be there.

This paper uses a modern version of this trick, but instead of just listening to sounds, it uses geometry.

The New Approach: Measuring Distances

The author, Nawapan Wattanawarichkul, introduces a new way to look at the problem using metrics (a fancy word for measuring distance).

Think of every possible pattern of numbers as a point on a giant map.

The Metric: The paper defines a special ruler that measures the "distance" between two different number patterns.
The Triangle Inequality: In geometry, the shortest path between two points is a straight line. If you go from Point A to Point B via Point C, the distance $A \to C \to B$ is always longer than or equal to the direct distance $A \to B$ .

The author sets up a triangle of three specific number patterns:

Pattern A: The pattern we are studying (our L-function).
Pattern B: A "mirror" version of that pattern.
Pattern C: A simple, known pattern (like the standard prime numbers).

By measuring the distances between these patterns, the author creates a mathematical inequality. If we assume a "ghost zero" exists in the dangerous zone, the math says the distance between the patterns would have to be negative. Since distance can't be negative, the ghost zero cannot exist.

The Two Main Innovations

The paper solves two specific problems that have been tricky for a long time:

1. Avoiding "Unproven" Assumptions
To use the old methods on these complex machines, mathematicians often had to assume a famous unproven theory called the Generalized Ramanujan Conjecture (GRC) was true. It's like trying to build a bridge assuming a specific type of steel exists, even though we haven't found it yet.

The Fix: This paper changes the "ingredients" of the measurement. Instead of measuring the raw patterns, it measures the rate of change (the derivative) of the patterns. This shift allows the author to prove the safe zones exist without needing to assume the GRC is true. It's like building the bridge using only materials we know for a fact exist.

2. Handling "Rough Terrain" (Ramified Primes)
Usually, these proofs work best when the number patterns are "smooth" (unramified). But sometimes, the patterns have "kinks" or "rough spots" (ramified primes).

The Fix: The author uses a concept called Positive Semi-Definite Families. Imagine a set of weights that are guaranteed to always balance on a scale in a positive direction. By arranging the L-functions into this specific family, the author can ignore the "rough spots" entirely. This leads to a cleaner, more precise proof that works even when the patterns are messy.

The Results: The "Safe Zones"

The paper proves three main theorems, which essentially say:

Theorem 1: For a single complex machine $L(s, \pi)$ , there is a wide safe zone where it never hits zero, unless it's a special "self-dual" machine that might have one single zero right on the edge.
The Theorem 2: If you combine two different machines ( $L(s, \pi \times \pi')$ ), and one of them is self-dual, there is a guaranteed safe zone.
The Theorem 3: If the combined machine has a special symmetry ( $L(s, \pi \times \pi') = L(s, \tilde{\pi} \times \tilde{\pi}')$ ), there is a safe zone, with at most one possible zero on the edge.

Why This Matters (According to the Paper)

The paper emphasizes that the "width" of these safe zones is very important. The author provides explicit constants, meaning they didn't just say "a safe zone exists"; they calculated exactly how wide it is.

This precision is crucial for other mathematicians who are trying to solve problems like the Sato-Tate conjecture (predicting how prime numbers behave in specific sequences). The better we know the size of the "safe zone," the more accurate our predictions about prime numbers can be.

What This Paper Does Not Do

It is important to note what this paper is not about:

It does not prove the Riemann Hypothesis (the ultimate goal of finding all zeros).
It does not apply these results to physics, engineering, or medicine.
It does not solve the problem for all possible combinations of machines; it specifically requires that at least one of the machines involved has a special "self-dual" symmetry. If you try to mix two completely random, non-symmetric machines, the paper admits that this specific geometric method hits a wall and cannot currently prove a safe zone.

Summary Analogy

Imagine you are a lighthouse keeper trying to ensure no ships (zeros) crash into the rocks (the critical line).

Old Method: You used a telescope that worked well only if you assumed the fog would clear up (GRC).
This Paper: You built a new radar system (the metric approach) that works perfectly even in thick fog. It uses the geometry of the waves themselves to prove that, in a specific wide area of the ocean, no ship can possibly be there. It's a more robust, self-contained way of keeping the waters safe.

Technical Summary: A Metric Approach to Zero-Free Regions for L-Functions

Problem Statement
The paper addresses the problem of establishing classical zero-free regions for standard $L$ -functions $L(s, \pi)$ and Rankin–Selberg $L$ -functions $L(s, \pi \times \pi')$ , where $\pi$ and $\pi'$ are cuspidal automorphic representations of $GL(m)$ and $GL(m')$ over a number field $F$ . Specifically, the goal is to prove that these functions have no zeros in a region of the form $\sigma \geq 1 - c/\log(C(\dots))$ , except possibly for a single simple real zero (Siegel zero) under specific self-duality conditions.

A primary challenge in extending classical proofs (such as those by Hadamard and de la Vallée Poussin for the Riemann zeta function) to the automorphic setting is the reliance on the Generalized Ramanujan Conjecture (GRC). Traditional approaches often require bounds on Dirichlet coefficients that are not yet fully established in the general case. Furthermore, previous results on Rankin–Selberg convolutions often lacked explicit constants or exhibited suboptimal dependence on the degrees $m$ and $m'$ .

Methodology
The author employs a "metric" approach inspired by the theory of "pretentious" multiplicative functions developed by Granville and Soundararajan, as formalized by Koukoulopoulos. The core strategy involves replacing the classical non-negative trigonometric inequalities (e.g., the $3-4-1$ argument) with a family of metrics defined on automorphic $L$ -functions.

Modified Metrics: Instead of using standard Dirichlet coefficients $\lambda_\pi(n)$ , the author defines a metric $D_\sigma$ based on the coefficients $a_\pi(n)$ of the logarithmic derivative $-(L'/L)(s, \pi)$ . This shift is crucial because $a_\pi(p) = \lambda_\pi(p)$ for primes, but the logarithmic derivative formulation allows the author to bypass the need for unproven progress toward the GRC regarding prime powers.
Triangle Inequalities: The proof architecture relies on the triangle inequality $D_\sigma(x_1, x_2) \leq D_\sigma(x_1, x_3) + D_\sigma(x_3, x_2)$ . By squaring this inequality and selecting specific elements (representations $\pi, \pi'$ , shifts $\gamma$ , and signs $\delta$ ), the author constructs non-negative linear combinations of logarithmic derivatives of $L$ -functions. This construction serves as an automorphic analogue of the classical trigonometric identities used to bound zero-free regions.
Handling Ramified Primes:
- Initial Approach (Section 4): When the level $Nq_\pi = 1$ , the metric approach directly yields the desired inequalities.
- Refined Approach (Section 5): When $Nq_\pi > 1$ , ramified primes introduce error terms. To eliminate the dependence on the GRC required to bound these terms, the author utilizes the framework of "positive semi-definite families" of automorphic $L$ -functions introduced by Lichtman and Pascadi. By defining a modified quantity $D^*_\sigma$ based on the positive semi-definiteness of the family $(-(L'/L)(s, \pi \times \tilde{\pi}'))$ , the author constructs the necessary non-negative linear combinations without including ramified prime contributions.

Key Results
The paper establishes the following theorems with explicit constants and uniform dependence on the degrees $m$ and $m'$ :

Theorem 1.1 (Standard $L$ -functions): For $\pi \in \mathcal{F}^*_m$ , $L(s, \pi)$ has no zeros in the region $\sigma \geq 1 - \frac{1}{33(2m+3)\log(C(\pi)(3+|t|)^{m[F:\mathbb{Q}]})}$ , except possibly for one simple real zero if $\pi$ is self-dual. If $\pi$ is self-dual, the exponent of $(3+|t|)$ can be reduced to $[F:\mathbb{Q}]$ .
Theorem 1.2 (Rankin–Selberg, Non-Self-Dual $\pi$ ): If $\pi \neq \tilde{\pi}$ and $\pi' = \tilde{\pi}'$ , then $L(s, \pi \times \pi') \neq 0$ in the region $\sigma \geq 1 - \frac{1}{28(m+m')\log(C(\pi)C(\pi')(3+|t|)^{m[F:\mathbb{Q}]})}$ .
Theorem 1.3 (Rankin–Selberg, Self-Dual Condition): If $L(s, \pi \times \pi') = L(s, \tilde{\pi} \times \tilde{\pi}')$ , then $L(s, \pi \times \pi')$ has at most one simple real zero in the region $\sigma \geq 1 - \frac{1}{66(m+m')\log(C(\pi)C(\pi')(3+|t|)^{m[F:\mathbb{Q}]})}$ .

Limitations and Structural Barriers
The paper explicitly analyzes the limitations of this metric approach. It demonstrates that if $\pi$ , $\pi'$ , and $L(s, \pi \times \pi')$ are all non-self-dual, the metric framework cannot produce a classical zero-free region. In this asymmetric setting, any linear combination derived from the triangle inequality satisfies $c_P \geq c_Z$ (where $c_P$ is the pole contribution and $c_Z$ is the zero contribution), failing the necessary criterion $0 \leq c_P < c_Z$ to derive a contradiction.

Significance and Novelty
The author claims the following contributions:

Independence from GRC: By utilizing the coefficients of the logarithmic derivative rather than the Dirichlet coefficients directly, the proofs avoid the need for progress toward the Generalized Ramanujan Conjecture.
Explicit Constants and Uniformity: The results provide explicit constants (e.g., 33, 28, 66) and improve upon or match the uniformity in $m$ and $m'$ compared to prior works by Brumley, Goldfeld, Li, Humphries, and Thorner.
New Zero-Free Regions: Theorem 1.3 is presented as a new result for the general case where $L(s, \pi \times \pi')$ is self-dual but $\pi$ and $\pi'$ are not necessarily self-dual individually (except in previously known special cases).
Methodological Alignment: The approach brings the study of standard and Rankin–Selberg $L$ -functions in line with the classical Hadamard–de la Vallée Poussin argument, offering a "metric" interpretation of the $3-4-1$ argument that recovers Mertens' result and extends it to the automorphic setting.
Correction of Oversight: The paper notes that it addresses a potential oversight in previous literature (specifically [11, Lemma 5.9]) regarding the contribution of poles at $s=1+i\gamma$ for $\gamma \neq 0$ , providing a more complete analysis in Section 4.1.2.

The paper concludes that while the metric approach is powerful under self-duality assumptions, the extension to general non-self-dual Rankin–Selberg $L$ -functions remains an open problem, suggesting that the positive semi-definite framework of Lichtman and Pascadi may offer a broader, though currently insufficient, path forward.